A175003 Triangle read by rows demonstrating Euler's pentagonal theorem for partition numbers.

1, 1, 1, 2, 1, 3, 2, 5, 3, -1, 7, 5, -1, 11, 7, -2, -1, 15, 11, -3, -1, 22, 15, -5, -2, 30, 22, -7, -3, 42, 30, -11, -5, 56, 42, -15, -7, 1, 77, 56, -22, -11, 1, 101, 77, -30, -15, 2, 135, 101, -42, -22, 3, 1, 176, 135, -56, -30, 5, 1, 231, 176, -77, -42, 7, 2

Offset

1,4

Comments

Row sums = A000041 starting with offset 1.

Sum of n-th row terms = leftmost term of next row, such that terms in each row demonstrate Euler's pentagonal theorem.

Let Q = triangle A027293 with partition numbers in each column.

Let M = a diagonalized variant of A080995 as the characteristic function of the generalized pentagonal numbers starting with offset 1: (1, 1, 0, 0, 1,...)

Sign the 1's: (++--++...) getting (1, 1, 0, 0, -1, 0, -1,...) which is the diagonal of matrix M, (as an infinite lower triangular matrix with the rest zeros).

Triangle A175003 = Q*M, with deleted zeros.

Column k starts at row A001318(k). - Omar E. Pol, Sep 21 2011

From Omar E. Pol, Apr 22 2014: (Start)

Row n has length A235963(n).

For Euler's pentagonal theorem for the sum of divisors see A238442.

Note that both of Euler's pentagonal theorems refer to generalized pentagonal numbers (A001318), not to pentagonal numbers (A000326). (End)

Links

Table of n, a(n) for n=1..66.

Formula

T(n,k) = A057077(k-1)*A000041(A195310(n,k)), n >= 1, k >= 1. - Omar E. Pol, Sep 21 2011

Example

Triangle begins:
    1;
    1,   1;
    2,   1;
    3,   2;
    5,   3,  -1;
    7,   5,  -1;
   11,   7,  -2,  -1;
   15,  11,  -3,  -1;
   22,  15,  -5,  -2;
   30,  22,  -7,  -3;
   42,  30, -11,  -5;
   56,  42, -15,  -7,   1;
   77,  56, -22, -11,   1;
  101,  77, -30, -15,   2;
  ...

Crossrefs

Cf. A000041, A080995, A027293, A238442.

Sequence in context: A240867 A242363 A050360 * A176431 A363083 A348112

Adjacent sequences: A175000 A175001 A175002 * A175004 A175005 A175006

Keyword

tabf,sign

Author

Gary W. Adamson, Apr 03 2010

Extensions

Corrected and extended by Omar E. Pol, Feb 14 2013

Status

approved